# Jordan magma

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties

## Definition

A magma $(S,*)$ is termed a Jordan magma if it satisfies the following two conditions:

1. Commutativity: $\! x * y = y * x \ \forall \ x,y \in S$.
2. Jordan's identity: $\! (x * y) * (x * x) = x * (y * (x * x))$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian semigroup
Abelian group

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Commutative magma any two elements commute (by definition) |FULL LIST, MORE INFO
Flexible magma $x * (y * x) = (x * y) * x$ (via commutativity) Commutative magma|FULL LIST, MORE INFO
Magma in which cubes are well-defined $x * (x * x) = (x * x) * x$ (via commutativity, flexibility) Commutative magma, Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Magma in which cubes and fourth powers are well-defined $x^3$ well-defined, all parenthesizations of $x^4$ also equal Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Magma in which powers up to the fifth are well-defined $x^3, x^4, x^5$ all well-defined |FULL LIST, MORE INFO